Equivalence of \(L_ p\) stability and exponential stability for a class of nonlinear semigroups. (English) Zbl 0547.47041

Let S(t) be a strongly continuous semigroup of bounded linear operators on a Banach space Y. A. Pazy has shown that if \(p\geq 1\) the p- integrability over [0,\(\infty)\) of S(t)y for any y in Y is equivalent to the exponential stability of S(t). R. Datko has then extended this to two-parameter semi-groups of bounded linear operators. In this paper a similar equivalence of \(L_ p\)-stability with \(p>0\) and exponential stability is shown for a certain class of nonlinear semigroups. It contains semilinear deterministic and stochastic evolution equations with Lipschitz nonlinearities. A possible application of this equivalence to obtain weaker sufficient conditions for exponential stability is discussed by a simple example.


47H20 Semigroups of nonlinear operators
93D20 Asymptotic stability in control theory
93E15 Stochastic stability in control theory
Full Text: DOI


[1] Curtain, R.F.; Pritchard, A.J., Infinite dimensional linear systems theory, () · Zbl 0352.49003
[2] Daleckii, Ju.L.; Krein, M.G., Stability of solutions of differential equations in Banach spaces, Transl. math. monogr., 43, (1974)
[3] Datko, R., Extending a theorem of A. M. Liapunov to Hilbert space, J. math. analysis applic., 32, 610-616, (1970) · Zbl 0211.16802
[4] Datko, T., Uniform asymptotic stability of evolutionary processes in a Banach space, SIAM J. math. analysis, 3, 428-445, (1972) · Zbl 0241.34071
[5] Hahn, W., Stability of motion, (1967), Springer New York · Zbl 0189.38503
[6] Has’minskii, R.Z., Stochastic stability of differential equations, (1980), Sijthoff & Noordhoff Alphen ann den Rijn · Zbl 0276.60059
[7] Haussmann, U.G., Asymptotic stability of the linear ito equation in infinite dimensions, J. math. analysis applic., 65, 219-235, (1978) · Zbl 0385.93051
[8] Ichikawa, A., Dynamic programming approach to stochastic evolution equations, SIAM J. control optim., 17, 152-174, (1979) · Zbl 0434.93069
[9] Ichikawa, A., Stability and optimal control of stochastic evolution equations, (), Chapter 5 · Zbl 0422.93102
[10] Ichikawa, A., Stability of semilinear stochastic evolution equations, J. math. analysis applic., 90, 12-44, (1982) · Zbl 0497.93055
[11] {\scIchikawa} A., Semilinear stochastic evolution equations: Boundedness, stability and invariant measures, Stochastics (to appear). · Zbl 0538.60068
[12] Pazy, A., On the applicability of Lyapunov’s theorem in Hilbert space, SIAM J. math. analysis, 3, 291-294, (1972) · Zbl 0242.47028
[13] Pritchard, A.J.; Zabczyk, J., Stability and stabilizability of infinite dimensional systems, SIAM rev., 23, 25-52, (1981) · Zbl 0452.93029
[14] Tanabe, H., Equations of evolution, (1979), Pitman London
[15] Wonham, W.M., Random differential equations in control theory, (), 131-212 · Zbl 0235.93025
[16] Zabczyk, J., Remarks on the control of discrete-time distributed parameter systems, SIAM J. control optim., 12, 721-735, (1974) · Zbl 0254.93027
[17] Zabczyk, J., On stability of infinite dimensional stochastic systems, (), 273-281
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.